Introduction to Real Analysis Dr. Weihu Hong Clayton State University 8/27/2009

Countable and Uncountable sets Equivalent sets: two sets A and B are said to be equivalent ， denoted A~B, if there exists a one-to-one function of A onto B. The notion of equivalence of sets is an equivalence relation (reflexive, symmetric, and transitive). Definition of finite, infinite, countable, uncountable, and at most countable: for each positive integer n, let If A is a set, we say: (a) A is finite if A ~ for some n, or A = Ø. (b) A is infinite if A is not finite. (c) A is countable if A ~ N. (d) A is uncountable if A is neither finite nor countable. (e) A is at most countable if A is finite or countable.

Examples of countable sets N is countable S = the set of positive integers that are perfect squares is countable Z is countable N × N is countable Q is countable E = the set of even integers is countable O = the set of odd integers is countable

Definition of Sequences If A is a set, by a sequence in A we mean a function from N into A, that is, f: N  A. For each n єN,. Then is called the nth term of the sequence f. If A is countable set, then there exists a one-to-one function f from N onto A. Thus A = Range f = The sequence f is called an enumeration of the set A; i.e.,

Theorem Every infinite subset of a countable set is countable. Proof: Let A be a countable set and let be an enumeration of A. Suppose E is an infinite subset of A. Then each x єE is of the form for some nєN. We can write E as a sequence in the natural order as in A, i.e., Define function f: N  E by Then f is one- to-one and onto. Thus E is countable.

Theorem If f maps N onto A, then A is at most countable. Proof: If A is finite, the A is at most countable. Suppose A is infinite. Since f maps N onto A, each a єA is an image of some nєN, that is, f(n) = a. Since the set is not empty, it follows from the well-ordering principle that it has a smallest integer, which denote by Consider the mapping a  of A into N. If then since f is a function,. Since A is infinite, is an infinite subset of N. Thus the mapping a  is a one-to-one mapping of A onto an infinite subset of N. It follows from Theorem that A is countable.

Indexed Families of Sets Let A and X be nonempty sets. An indexed family of subsets of X with index set A is a function from A to into P (X), denote by Examples: For each x є(0,1), let Then is an indexed family of subsets of Q. The sequence

Operations of indexed families of subsets of X Suppose is an indexed family of subsets of X. The union and intersection of the family are defined respectively

Distributive and De Morgan’s laws

The Countability of Q Theorem If is a sequence of countable sets and then S is countable. Proof: Since is countable for each n єN, we can write Since is an infinite subset of S, the set S itself is infinite. Consider the function h: N×N  S by The function h (may not be one-to-one) is a mapping of N×N onto S. Thus since N×N ~ N, there exists a mapping of N onto S. It follows from Theorem that the set S is countable.

Corollary Q is countable. Proof: Since each rational can be written as a fraction of two integers, for each m єN, let Then is countable, and it follows from Theorem that the set Q is countable.

Theorem The closed interval [0,1] is uncountable. Proof: (Cantor) Since there are infinitely many rational numbers in [0,1],the set is not finite. To prove that it is uncountable, we only need to show that it is not countable. To this end, we will prove that every countable subset of [0,1] is a proper subset of [0,1]. Thus [0,1] cannot be countable. Let be a countable subset of [0,1]. Then each has a decimal expansion

Continuation of the proof: Define a new number as follows Therefore, E is a proper subset of [0,1].